The word average can conceal a choice about which observations count most. Equal weighting, weighting by the amount of material and weighting by measurement precision can all produce mathematically valid calculations. They are not interchangeable because they can target different quantities or rely on different assumptions.
Make the weights visible
A weighted mean multiplies each value by its weight, adds those products and divides by the sum of the weights. Equal weights reduce that equation to the ordinary arithmetic mean.
For two illustrative values, 90% and 98%, equal weights give 94%. Weights of 1 and 9 give (90 × 1 + 98 × 9) ÷ 10 = 97.2%. Multiplying both weights by ten would leave the answer unchanged.
| Weight represents | Possible question |
|---|---|
| One per sampled unit | What is the mean across these units? |
| Material mass | What is the combined material mass fraction? |
| Inverse measurement variance | What common value is estimated by these measurements? |
The table describes different purposes. It does not establish that the available samples represent a whole lot or that measurements from different methods estimate one common value.
Distinguish a mean of percentages from combined composition
Imagine two original hypothetical portions: 1 g with a target mass fraction of 90%, and 9 g with a fraction of 98%. Their target masses are 0.90 g and 8.82 g. Together they contain 9.72 g target in 10 g total, giving 97.2%.
The equal-portion mean of 94% answers a different question. It treats the two portions as equally important observations regardless of their masses. It does not describe the composition of their hypothetical combination.
This calculation assumes compatible mass-fraction definitions and no losses or transformations on combining. It cannot be transferred directly to chromatographic area percentages, which do not automatically represent target mass fractions.
Understand what precision weighting assumes
A National Bureau of Standards paper on consensus values explains inverse-variance weighting for independent estimates of a common value and discusses the difficulty of estimating suitable variances. Smaller variance receives greater weight under that model.NBS / NIST — Consensus Values and Weighting Factors (opens in a new tab)
For an original example, two independent estimates are 10.0 mg and 10.4 mg, with standard uncertainties of 0.1 and 0.2 mg. Inverse squared uncertainties give weights of 100 and 25. The resulting weighted estimate is 10.08 mg, compared with the equal-weight mean of 10.20 mg.
That calculation assumes the uncertainties describe the relevant measurement variation and the estimates address the same quantity. A highly repeatable but biased method should not acquire unquestioned authority merely because its repeatability number is small.
Shared calibration effects also challenge the independence assumption. Treating correlated estimates as independent can overstate the information gained from combining them.
Read the average as part of its evidence
Ask for the individual results, weights, definition of the target quantity and treatment of missing or qualified values. The final mean alone cannot reveal whether the calculation weighted vials, grams, replicate counts or analytical precision.
A laboratory reporting more replicate readings should not automatically dominate a lot-level summary if those readings all came from one vial. More instrument observations and more independently sampled units are different information.
Retain the spread and relevant individual results alongside the mean. A weighted summary can conceal an unusual unit or a disagreement between methods even when its arithmetic is correct.
Where the weighting rationale is missing, describe the reported number as an unexplained summary. Recalculating alternatives can demonstrate sensitivity, but it does not establish which quantity the original author intended.
Sources and further detail
- NBS / NIST — Consensus Values and Weighting Factors (opens in a new tab)
Journal of Research of the National Bureau of Standards, 87(5), 1982; common-value model and inverse-variance weighting discussion read. All numerical examples here are original and their assumptions are stated.
Sources checked 19 September 2026. Worked examples are illustrative unless a supplied report is explicitly identified. This article has not undergone independent scientific peer review.